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Maths-

General

Easy

Question

If $f left parenthesis x right parenthesis equals fraction numerator g left parenthesis x right parenthesis plus g left parenthesis negative x right parenthesis over denominator 2 end fraction plus fraction numerator 2 over denominator left ceiling h left parenthesis x right parenthesis plus h left parenthesis negative x right parenthesis right square bracket to the power of negative 1 end exponent end fraction$ where g and are differentiable function, then $f to the power of straight prime left parenthesis 0 right parenthesis$ is

1
$\frac{1}{2}$
$\frac{3}{2}$
0

The correct answer is: 0

Given
$table attributes columnspacing 1em end attributes row cell straight f left parenthesis straight x right parenthesis equals fraction numerator straight g left parenthesis straight x right parenthesis plus straight g left parenthesis negative straight x right parenthesis over denominator 2 end fraction plus fraction numerator 2 over denominator left ceiling straight h left parenthesis straight x right parenthesis plus straight h left parenthesis negative straight x right parenthesis right ceiling to the power of negative 1 end exponent end fraction end cell row cell not stretchy rightwards double arrow left parenthesis straight x right parenthesis equals fraction numerator straight g left parenthesis straight x right parenthesis plus straight g left parenthesis negative straight x right parenthesis over denominator 2 end fraction plus 2 left square bracket straight h left parenthesis straight x right parenthesis plus straight h left parenthesis negative straight x right parenthesis right square bracket end cell end table$
On differentiating w.r.t. we get
$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell straight f to the power of straight prime left parenthesis straight x right parenthesis equals fraction numerator straight g to the power of straight prime left parenthesis straight x right parenthesis minus straight g to the power of straight prime left parenthesis negative straight x right parenthesis over denominator 2 end fraction plus 2 open square brackets straight h to the power of straight prime left parenthesis straight x right parenthesis minus straight h to the power of straight prime left parenthesis negative straight x right parenthesis close square brackets end cell row cell therefore straight f to the power of straight prime left parenthesis 0 right parenthesis equals fraction numerator straight g to the power of straight prime left parenthesis 0 right parenthesis minus straight g to the power of straight prime left parenthesis 0 right parenthesis over denominator 2 end fraction plus 2 open square brackets straight h to the power of straight prime left parenthesis 0 right parenthesis minus straight h to the power of straight prime left parenthesis 0 right parenthesis close square brackets end cell row cell equals 0 end cell end table$

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